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Quasivarieties of cancellative commutative binary modes. (English) Zbl 1092.08005

For terminology and background, see [A. B. Romanowska and J. D. H. Smith, Modal theory. Heldermann, Berlin (1985; Zbl 0553.08001); Modes. World Scientific, Singapore (2002; Zbl 1012.08001)].
The paper under review initiates an investigation of the lattice of all quasivarieties of commutative binary modes by studying commutative quasigroup modes and cancellative commutative binary modes. A description of the lattice of all quasivarieties of affine spaces over a principal ideal domain (PID) is provided. This lattice is isomorphic to the lattice of quasivarieties of modules over a PID. This leads to a description of the lattice of quasivarieties of commutative quasigroup modes, since this variety is equivalent to the variety of affine spaces over the PID D of rational dyadic numbers. Then, quasivarieties of cancellative commutative binary modes are discussed. Each quasivariety of affines spaces over D determines a quasivariety of cancellative binary modes. It is shown that these are precisely all the quasivarietes of cancellative commutative binary modes.

MSC:

08C15 Quasivarieties
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References:

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